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Quantum Particle Swarm Optimization Algorithm Based on Dynamic Adaptive .... Jing Huo 325
Here, for
1 2
i i
pbest gbest
p
, the formula 12 shows that when 1 1
, under the
condition that
t
,
1 2
i
pbest gbest
x t
, i.e. the algorithm convergence.
i
p
is referred to the local attractor of the particle
i
. If QPSO algorithm convergence can be guaranteed, it requires that each particle should converge to its own local attractor
i
p
:
1
i i
p pbest
gbest
13 Here,
1
From formula 13, we can see that the local attractor
i
p
is located at the super rectangle with the individual best position
i
pbest
and the group best position
gbest
as the vertex, and the position of the local attractor
i
p
changes along with the change of
i
pbest
. When the algorithm converges, the particle swarm also converges to the local attractor
i
p
, at that time, the particle individual best position
i
pbest
, the group best position
gbest
and the local attractor
i
p
overlap in one point. Therefore, it is assumed that during the algorithm iteration process, there exists the attract potential field in some form at the local attractor
i
p
, and all particles in the population is attracted by
i
p
,
and approach gradually to
i
p
with the algorithm iteration and eventually overlap with
i
p
,
that is also the reason why particle swarm is able to maintain the aggregation [14].
3.3. Algorithm description
For classical mechanics, the flying track of the particle is fixed, but for the quantum mechanics, we can see from the Heisenberg uncertainty principle that for a particle, its position
and speed cannot be determined at the same time, and the track makes no sense. Therefore, if the particle in PSO algorithm has the quantum behavior in the quantum mechanics, then the
PSO algorithm will work in different ways. The algorithm flow chart of this article is as follows [15],[16] as Figure 2.
For quantum mechanics, the particle state is described by the wave function
, X t
which is the complex function of coordinate and time, in which,
, , X
x y z
is the position vector of the particle in three-dimensional space. The wave function’s physical meaning is: the square
of the wave function module is the possibility density when one particle occurs at a certain point
X
in the space at the time point
t
, i.e.:
2
dxdydz Qdxdydz
14 In which, Q is the probability density function. Probability distribution density function
satisfies the normalization condition:
2
1 dxdydz
Qdxdydz
15 Suppose the particle swarm system is a quantum system, and each particle has the
quantum behavior, and the wave function is used to describe the particle state. According to the analysis of the convergence behavior of particles in PSO algorithm, there must be attract
potential in some form centered by
i
p
.
So, potential well can be set up at
i
p
and its potential energy function is expressed as:
i
V x X
p Y
16
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 1, March 2015 : 321 – 330
326
Figure 2. Algorithm flow chart In which,
i
Y X
p
, m is the quality of the particle, so Hamiltonian operator of such question is as follows:
2 2
2
2 ћ d
H Y
m dY
17 The particle’s time-independent Schrodinger equation in
potential well is:
2 2
2
2 2
d m
E Y
Y ћ
18 In which,
E
is the particle energy. By solving the corresponding wave function of this equation, we can get:
2
1 ,
Y L
ћ Y
e L
m L
19 Its possibility density function Q is:
2 2
1
Y L
Q Y Y
e L
20 Possibility distribution function
F
is:
Initialization population Initialization particle historical best
and global historical best
Update all particles in the population Evaluate the particle fitness function
in the population Update the particle historical best
Update the global historical best of particle swarm
Meet end condition? Yes
Output optimal solution No
TELKOMNIKA ISSN: 1693-6930
Quantum Particle Swarm Optimization Algorithm Based on Dynamic Adaptive .... Jing Huo 327
2
1
Y L
F Y e
21 Quantum state function
Y
only calculates the probability density function
2
Y
or
Q Y when the particle occurs at
i
p
.
Monte Carlo stochastic simulation can be adopted to determine the particle position:
1 1
ln 2
i i
x p
u
22 In which,
is the random number among 0,1, and
L
is the characteristic length of
potential well. In order to make the particle position change along with the time and also able to converge, the characteristic length in formula 22 must also change along with the time, namely
L L t
. In this way formula 22 can be rewritten as:
1 1
ln 2
i i
i i
L t x t
p u t
23 From formula 22, we can see that
L
is the search scope of particles, and the larger
L
value is, the larger the particle search scope. However, if the
L
is too large, it will lead the entire particle swarm to diverge and lower the particle swarm convergence speed and ability, if
L
value is too small, it will lead the premature convergence of particle swarm and also fall into the local best. The following two methods can be adopted to evaluate
L
value choice: The first method is:
2
i i
i
L t p t
x t
24 Then, for
1 t
generation, the position evolution equation of number
i
in j dimension can turn into:
1 1
ln
ij ij
ij ij
ij
x t
p t
p t
x t
u t
25 The second method is to introduce the mean best position which is defined as the
average of all individual particles’ best position, namely:
1 2
1 1
1 1
1 1
1 1
, ,
,
M M
M M
i i
besti iD
i i
i i
mbest pbest
pbest p
pbest M
M M
M
26
Then,
L s
evaluation method may be changed into:
2
i i
L t mbest
x t
27 Corresponding particle evolution formula is changed into:
1 1
ln
ij ij
j ij
ij
x t
p t
mbest x
t u
t
28
In which, is called the contraction-expansion coefficient, and it is the only parameter
except the group size and iteration number in the QPSO algorithm. is used to control the
convergence speed of particles, when 1.75
, the algorithm can guarantee the convergence,
and generally the linear decreasing gradient from 1 to 0.5 is selected.
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 1, March 2015 : 321 – 330
328
4. Test and Analysis 4.1. Standard test function
